pow!

Author: rrbutani

src/math/pow.rs

@@ -0,0 +1,395 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/* pow(x,y) return x**y
+ *
+ *                    n
+ * Method:  Let x =  2   * (1+f)
+ *      1. Compute and return log2(x) in two pieces:
+ *              log2(x) = w1 + w2,
+ *         where w1 has 53-24 = 29 bit trailing zeros.
+ *      2. Perform y*log2(x) = n+y' by simulating muti-precision
+ *         arithmetic, where |y'|<=0.5.
+ *      3. Return x**y = 2**n*exp(y'*log2)
+ *
+ * Special cases:
+ *      1.  (anything) ** 0  is 1
+ *      2.  1 ** (anything)  is 1
+ *      3.  (anything except 1) ** NAN is NAN
+ *      4.  NAN ** (anything except 0) is NAN
+ *      5.  +-(|x| > 1) **  +INF is +INF
+ *      6.  +-(|x| > 1) **  -INF is +0
+ *      7.  +-(|x| < 1) **  +INF is +0
+ *      8.  +-(|x| < 1) **  -INF is +INF
+ *      9.  -1          ** +-INF is 1
+ *      10. +0 ** (+anything except 0, NAN)               is +0
+ *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
+ *      12. +0 ** (-anything except 0, NAN)               is +INF, raise divbyzero
+ *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF, raise divbyzero
+ *      14. -0 ** (+odd integer) is -0
+ *      15. -0 ** (-odd integer) is -INF, raise divbyzero
+ *      16. +INF ** (+anything except 0,NAN) is +INF
+ *      17. +INF ** (-anything except 0,NAN) is +0
+ *      18. -INF ** (+odd integer) is -INF
+ *      19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
+ *      20. (anything) ** 1 is (anything)
+ *      21. (anything) ** -1 is 1/(anything)
+ *      22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
+ *      23. (-anything except 0 and inf) ** (non-integer) is NAN
+ *
+ * Accuracy:
+ *      pow(x,y) returns x**y nearly rounded. In particular
+ *                      pow(integer,integer)
+ *      always returns the correct integer provided it is
+ *      representable.
+ *
+ * Constants :
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+
+// #include "libm.h"
+
+/* Concerns:
+ *  - Some constants are shared; DRY?
+ *  - FLT_EVAL_METHOD: the others sidestep this (epsilon or just always true in the case of hypot (#71))
+ */
+
+use super::{fabs, scalbn, sqrt, with_set_low_word, with_set_high_word, get_high_word};
+
+const BP:   [f64; 2] = [1.0, 1.5];
+const DP_H: [f64; 2] = [0.0, 5.84962487220764160156e-01]; /* 0x3fe2b803_40000000 */
+const DP_L: [f64; 2] = [0.0, 1.35003920212974897128e-08]; /* 0x3E4CFDEB, 0x43CFD006 */
+const TWO53: f64     =  9007199254740992.0; /* 0x43400000_00000000 */
+const HUGE:  f64     =  1.0e300;
+const TINY:  f64     =  1.0e-300;
+
+// poly coefs for (3/2)*(log(x)-2s-2/3*s**3:
+const L1: f64 =  5.99999999999994648725e-01; /* 0x3fe33333_33333303 */
+const L2: f64 =  4.28571428578550184252e-01; /* 0x3fdb6db6_db6fabff */
+const L3: f64 =  3.33333329818377432918e-01; /* 0x3fd55555_518f264d */
+const L4: f64 =  2.72728123808534006489e-01; /* 0x3fd17460_a91d4101 */
+const L5: f64 =  2.30660745775561754067e-01; /* 0x3fcd864a_93c9db65 */
+const L6: f64 =  2.06975017800338417784e-01; /* 0x3fca7e28_4a454eef */
+const P1: f64 =  1.66666666666666019037e-01; /* 0x3fc55555_5555553e */
+const P2: f64 = -2.77777777770155933842e-03; /* 0xbf66c16c_16bebd93 */
+const P3: f64 =  6.61375632143793436117e-05; /* 0x3f11566a_af25de2c */
+const P4: f64 = -1.65339022054652515390e-06; /* 0xbebbbd41_c5d26bf1 */
+const P5: f64 =  4.13813679705723846039e-08; /* 0x3e663769_72bea4d0 */
+const LG2:     f64 =  6.93147180559945286227e-01; /* 0x3fe62e42_fefa39ef */
+const LG2_H:   f64 =  6.93147182464599609375e-01; /* 0x3fe62e43_00000000 */
+const LG2_L:   f64 = -1.90465429995776804525e-09; /* 0xbe205c61_0ca86c39 */
+const OVT:     f64 =  8.0085662595372944372e-017; /* -(1024-log2(ovfl+.5ulp)) */
+const CP:      f64 =  9.61796693925975554329e-01; /* 0x3feec709_dc3a03fd =2/(3ln2) */
+const CP_H:    f64 =  9.61796700954437255859e-01; /* 0x3feec709_e0000000 =(float)cp */
+const CP_L:    f64 = -7.02846165095275826516e-09; /* 0xbe3e2fe0_145b01f5 =tail of cp_h*/
+const IVLN2:   f64 =  1.44269504088896338700e+00; /* 0x3ff71547_652b82fe =1/ln2 */
+const IVLN2_H: f64 =  1.44269502162933349609e+00; /* 0x3ff71547_60000000 =24b 1/ln2*/
+const IVLN2_L: f64 =  1.92596299112661746887e-08; /* 0x3e54ae0b_f85ddf44 =1/ln2 tail*/
+
+#[inline]
+pub fn pow(x: f64, y: f64) -> f64 {
+    let t1: f64;
+    let t2: f64;
+
+    let (hx, lx): (i32, u32) = ((x.to_bits() >> 32) as i32, x.to_bits() as u32);
+    let (hy, ly): (i32, u32) = ((y.to_bits() >> 32) as i32, y.to_bits() as u32);
+    
+    let mut ix: i32 = (hx & 0x7fffffff) as i32;
+    let iy: i32 = (hy & 0x7fffffff) as i32;
+
+    /* x**0 = 1, even if x is NaN */
+    if ((iy as u32) | ly) == 0 {
+        return 1.0;
+    }
+
+    /* 1**y = 1, even if y is NaN */
+    if hx == 0x3ff00000 && lx == 0 {
+        return 1.0;
+    }
+
+    /* NaN if either arg is NaN */
+    if ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
+        iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0) {
+            return x + y;
+    }
+
+    /* determine if y is an odd int when x < 0
+     * yisint = 0       ... y is not an integer
+     * yisint = 1       ... y is an odd int
+     * yisint = 2       ... y is an even int
+     */
+    let mut yisint: i32 = 0;
+    let mut k: i32;
+    let mut j: i32;
+    if hx < 0 {
+        if iy >= 0x43400000 {
+            yisint = 2; /* even integer y */
+        } else if iy >= 0x3ff00000 {
+            k = (iy >> 20) - 0x3ff; /* exponent */
+            
+            if k > 20 {
+                j = (ly >> (52 - k)) as i32;
+                
+                if (j << (52 - k)) == (ly as i32) {
+                    yisint = 2 - (j & 1);
+                }
+            } else if ly == 0 {
+                j = iy >> (20 - k);
+                
+                if (j << (20 - k)) == iy {
+                    yisint = 2 - (j & 1);
+                }
+            }
+        }
+    }
+
+    if ly == 0 {
+        /* special value of y */
+        if iy == 0x7ff00000 {
+            /* y is +-inf */    
+            return if ((ix - 0x3ff00000) | (lx as i32)) == 0 {
+                /* (-1)**+-inf is 1 */
+                1.0
+            } else if ix >= 0x3ff00000 {
+                /* (|x|>1)**+-inf = inf,0 */
+                if hy >= 0 { y } else { 0.0 }
+            } else {
+                /* (|x|<1)**+-inf = 0,inf */
+                if hy >= 0 { 0.0 } else { -y }
+            };
+        }
+
+        if iy == 0x3ff00000 {
+            /* y is +-1 */
+            return if hy >= 0 { x } else { 1.0 / x };
+        }
+
+        if hy == 0x40000000 {
+            /* y is 2 */
+            return x * x;
+        }
+
+        if hy == 0x3fe00000 {
+            /* y is 0.5 */
+            if hx >= 0 {
+                /* x >= +0 */
+                return sqrt(x);
+            }
+        }
+    }
+
+    let mut ax: f64 = fabs(x);
+    if lx == 0 {
+        /* special value of x */
+        if ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000 {
+            /* x is +-0,+-inf,+-1 */
+            let mut z: f64 = ax;
+            
+            if hy < 0 {
+                /* z = (1/|x|) */
+                z = 1.0 / z;
+            }
+
+            if hx < 0 {
+                if ((ix-0x3ff00000)|yisint) == 0 {
+                    z = (z - z) / (z - z); /* (-1)**non-int is NaN */
+                } else if yisint == 1 {
+                    z = -z; /* (x<0)**odd = -(|x|**odd) */
+                }
+            }
+
+            return z;
+        }
+    }
+
+    let mut s: f64 = 1.0; /* sign of result */
+    if hx < 0 {
+        if yisint == 0 {
+            /* (x<0)**(non-int) is NaN */
+            return (x - x) / (x - x);
+        }
+
+        if yisint == 1 {
+            /* (x<0)**(odd int) */
+            s = -1.0;
+        }
+    }
+
+    /* |y| is HUGE */
+    if iy > 0x41e00000 {
+        /* if |y| > 2**31 */
+        if iy > 0x43f00000 {
+            /* if |y| > 2**64, must o/uflow */
+            if ix <= 0x3fefffff {
+                return if hy < 0 { HUGE * HUGE } else { TINY * TINY };
+            }
+
+            if ix >= 0x3ff00000 {
+                return if hy > 0 { HUGE * HUGE } else { TINY * TINY };
+            }
+        }
+
+        /* over/underflow if x is not close to one */
+        if ix < 0x3fefffff {
+            return if hy < 0 { s * HUGE * HUGE } else { s * TINY * TINY };
+        }
+        if ix > 0x3ff00000 {
+            return if hy > 0 { s * HUGE * HUGE } else { s * TINY * TINY };
+        }
+
+        /* now |1-x| is TINY <= 2**-20, suffice to compute
+           log(x) by x-x^2/2+x^3/3-x^4/4 */
+        let t: f64 = ax - 1.0;         /* t has 20 trailing zeros */
+        let w: f64 = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
+        let u: f64 = IVLN2_H * t;  /* ivln2_h has 21 sig. bits */
+        let v: f64 = t * IVLN2_L - w * IVLN2;
+        t1 = with_set_low_word(u + v, 0);
+        t2 = v - (t1 - u);
+    } else {
+        // double ss,s2,s_h,s_l,t_h,t_l;
+        let mut n: i32 = 0;
+
+        if ix < 0x00100000 {
+            /* take care subnormal number */
+            ax *= TWO53;
+            n  -= 53;
+            ix  = get_high_word(ax) as i32;
+        }
+
+        n += (ix >> 20) - 0x3ff;
+        j = ix & 0x000fffff;
+
+        /* determine interval */
+        let k: i32;
+        ix = j | 0x3ff00000;   /* normalize ix */
+        if j <= 0x3988E {
+            /* |x|<sqrt(3/2) */
+            k = 0;
+        }
+        else if j < 0xBB67A {
+            /* |x|<sqrt(3)   */
+            k = 1;
+        } else {
+            k = 0;
+            n += 1;
+            ix -= 0x00100000;
+        }
+        ax = with_set_high_word(ax, ix as u32);
+
+        /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
+        let u: f64 = ax - BP[k as usize]; /* bp[0]=1.0, bp[1]=1.5 */
+        let v: f64 = 1.0 / (ax + BP[k as usize]);
+        let ss: f64  = u * v;
+        let s_h = with_set_low_word(ss, 0);
+
+        /* t_h=ax+bp[k] High */
+        let t_h: f64 = with_set_high_word(0.0,
+                ((ix as u32 >> 1) | 0x20000000) + 0x00080000 + ((k as u32) << 18));
+        let t_l: f64 = ax - (t_h - BP[k as usize]);
+        let s_l: f64 = v * ((u - s_h * t_h) - s_h * t_l);
+
+        /* compute log(ax) */
+        let s2: f64 = ss * ss;
+        let mut r: f64 = s2 * s2 * (L1 + s2 * (L2 + s2 *(L3 + s2 *(L4 + s2 *(L5 + s2 * L6)))));
+        r += s_l * (s_h + ss);
+        let s2: f64 = s_h * s_h;
+        let t_h: f64 = with_set_low_word(3.0 + s2 + r, 0);
+        let t_l: f64 = r - ((t_h - 3.0) - s2);
+
+        /* u+v = ss*(1+...) */
+        let u: f64 = s_h * t_h;
+        let v: f64 = s_l * t_h + t_l * ss;
+
+        /* 2/(3log2)*(ss+...) */
+        let p_h: f64 = with_set_low_word(u + v, 0);
+        let p_l = v - (p_h-u);
+        let z_h: f64 = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */
+        let z_l: f64 = CP_L * p_h + p_l * CP + DP_L[k as usize];
+
+        /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
+        let t: f64 = n as f64;
+        t1 = with_set_low_word(((z_h + z_l) + DP_H[k as usize]) + t, 0);
+        t2 = z_l - (((t1 - t) - DP_H[k as usize]) - z_h);
+    }
+
+    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
+    let y1: f64  = with_set_low_word(y, 0);
+    let p_l: f64 = (y - y1) * t1 + y * t2;
+    let mut p_h: f64 = y1 * t1;
+    let z: f64   = p_l + p_h;
+    let mut j: i32 = (z.to_bits() >> 32) as i32;
+    let i: i32 = z.to_bits() as i32;
+    // let (j, i): (i32, i32) = ((z.to_bits() >> 32) as i32, z.to_bits() as i32);
+
+    if j >= 0x40900000 {
+        /* z >= 1024 */
+        if (j - 0x40900000) | i != 0 {
+            /* if z > 1024 */
+            return s * HUGE * HUGE; /* overflow */
+        }
+
+        if p_l + OVT > z - p_h {
+            return s * HUGE * HUGE; /* overflow */
+        }
+    } else if (j & 0x7fffffff) >= 0x4090cc00 {
+        /* z <= -1075 */
+        // FIXME: instead of abs(j) use unsigned j
+
+        if (((j as u32) - 0xc090cc00) | (i as u32)) != 0 {
+            /* z < -1075 */
+            return s * TINY * TINY; /* underflow */
+        }
+
+        if p_l <= z - p_h {
+            return s * TINY * TINY; /* underflow */
+        }
+    }
+
+    /* compute 2**(p_h+p_l) */
+    let i: i32 = j & (0x7fffffff as i32);
+    k = (i >> 20) - 0x3ff;
+    let mut n: i32 = 0;
+
+    if i > 0x3fe00000 {
+        /* if |z| > 0.5, set n = [z+0.5] */
+        n = j + (0x00100000 >> (k + 1));
+        k = ((n&0x7fffffff) >> 20) - 0x3ff; /* new k for n */
+        let t: f64 = with_set_high_word(0.0, (n & !(0x000fffff >> k)) as u32);
+        n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
+        if j < 0 {
+            n = -n;
+        }
+        p_h -= t;
+    }
+
+    let t: f64 = with_set_low_word(p_l + p_h, 0);
+    let u: f64 = t * LG2_H;
+    let v: f64 = (p_l - (t - p_h)) * LG2 + t * LG2_L;
+    let mut z: f64 = u + v;
+    let w: f64 = v - (z - u);
+    let t: f64 = z * z;
+    let t1: f64 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
+    let r: f64 = (z * t1) / (t1 - 2.0) - (w + z*w);
+    z = 1.0 - (r - z);
+    j = get_high_word(z) as i32;
+    j += n << 20;
+
+    if (j >> 20) <= 0 {
+        /* subnormal output */
+        z = scalbn(z,n);
+    } else {
+        z = with_set_high_word(z, j as u32);
+    }
+
+    return s*z;
+}